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# Effective atomic number

Effective atomic number has two different meanings: one that is the effective nuclear charge of an atom, and one that calculates the average atomic number for a compound or mixture of materials. Both are abbreviated Zeff.

## For an atom

The effective atomic number Zeff', (sometimes referred to as the effective nuclear charge) of an atom is the number of protons an electron in the element effectively 'sees' due to screening by inner-shell electrons. It is a measure of the electrostatic interaction between the negatively charged electrons and positively charged protons in the atom. One can view the electrons in an atom as being 'stacked' by energy outside the nucleus; the lowest energy electrons (such as the 1s and 2s electrons) occupy the space closest to the nucleus, and electrons of higher energy are located further from the nucleus.

The binding energy of an electron, or the energy needed to remove the electron from the atom, is a function of the electrostatic interaction between the negatively charged electrons and the positively charged nucleus. In Iron, atomic number 26, for instance, the nucleus contains 26 protons. The electrons that are closest to the nucleus will 'see' nearly all of them. However, electrons further away are screened from the nucleus by other electrons in between, and feel less electrostatic interaction as a result. The 1s electron of Iron (the closest one to the nucleus) sees an effective atomic number (number of protons) of 25. The reason why it is not 26 is because some of the electrons in the atom end up repelling the others, giving a net lower electrostatic interaction with the nucleus. One way of envisioning this effect is to imagine the 1s electron sitting on one side of the 26 protons in the nucleus, with another electron sitting on the other side; each electron will feel less than the attractive force of 26 protons because the other electron contributes a repelling force. The 4s electrons in Iron, which are furthest from the nucleus, feel an effective atomic number of only 5.43 because of the 25 electrons in between it and the nucleus screening the charge.

Effective atomic numbers are useful not only in understanding why electrons further from the nucleus are so much more weakly bound than those closer to the nucleus, but also because they can tell us when to use simplified methods of calculating other properties and interactions. For instance, Lithium, atomic number 3, has two electrons in the 1s shell and one in the 2s shell. Because the two 1s electrons screen the protons to give an effective atomic number for the 2s electron close to 1, we can treat this 2s valence electron with a hydrogenic model.

Mathematically, the effective atomic number Zeffcan be calculated using methods known as "self-consistent field" calculations, but in simplified situations is just taken as the atomic number minus the number of electrons between the nucleus and the electron being considered.

## For a compound or mixture

Effective atomic number is a term that is similar to atomic number but is used for compounds (e.g. water) and mixtures of different materials (such as tissue and bone) rather than for atoms. The effective atomic number is calculated by taking the fractional proportion of each atom in the compound and multiplying that by the atomic number of the atom. The formula for the effective atomic number, Zeff, is as follows:

$Z_{eff} = \sqrt[2.94]{f_{1} \times (Z_{1})^{2.94} + f_{2} \times (Z_{2})^{2.94} + f_{3} \times (Z_{3})^{2.94} + ...}$
where
fn is the fraction of the total number of electrons associated with each element, and
Zn is the atomic number of each element.

An example is that of water (H2O), made up of two hydrogen atoms (Z=1) and one oxygen atom (Z=8), the total number of electrons is 1+1+8 = 10, so the fraction of electrons for the two hydrogens is (2/10) and for the one oxygen is (8/10). So the Zeff for water is:

$Z_{eff} = \sqrt[2.94]{0.2 \times 1^{2.94} + 0.8 \times 8^{2.94}} = 7.42$

Effective atomic number is important for predicting how X-rays interact with a substance, as certain types of x-ray interactions depend on the atomic number.

## References

• Webelements
• Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles.
• R. C. Murty, "Effective atomic numbers of heterogeneous materials", Nature 207, 398-399 (24 July 1965)